The Certainty of Quantum Set Theory
Posted: Sun Mar 26, 2017 5:36 pmHere's my brief two-paragraph summary of quantum set theory (QST):
Birkhoff and von Neumann [11] introduced quantum logic in response to logical problems with the Copenhagen interpretation. Takeuti [8] formed the quantum logic into an introduction of QST. Eventually, Ozawa took the lead to develop QST into a feasible interpretation of quantum mechanics (QM) that coheres with the classical law of noncontradiction, predicate logic, and experimental physics [9, 10, 12, 13]. For example, QST defines quantum states with certainty instead of classical uncertainty.
In short, QST begins with a prior probability distribution of observables for a particular quantum state. This distribution looks similar to a corresponding Copenhagen probability distribution of observables for the quantum state, but QST assigns predicate logic to each observable in the prior set for the quantum state. For example, the existence of each observable in a particular quantum state is true or false. This results in a set of existing observables that completely defines the quantum state despite classically non-commuting observables such as momentum and position. Additionally, QST can define entangled states because it is a state-dependent theory instead of a particle-dependent theory. Furthermore, QST preserves the Copenhagen probability distribution for wave function collapse that results in probabilistic causality during the transition from one quantum state to the next. For instance, wave function collapse is the only element of uncertainty in QST.
References
8. Takeuti, G.: Quantum set theory. In: Beltrametti, E.G., van Fraassen, B.C. (eds.) Current Issues in Quantum Logic, pp. 303–322. Plenum, New York (1981)
9. Ozawa, M. Quantum reality and measurement: A quantum logical approach. Found. Phys. 41, 592–607 (2011)
10. Ozawa, M. Quantum set theory extending the standard probabilistic interpretation of quantum theory. New Generat. Comput. 34, 125–152 (2016)
11. Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)
12. Sulyok, G., Sponar, S., Demirel, B., Buscemi, F., Hall, M.J.W., Ozawa, M., Hasegawa, Y.: Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements. Phys. Rev. Lett. 115, 030401 (2015)
13. Demirel, B., Sponar, S., Sulyok, G., Ozawa, M., Hasegawa, Y.: Experimental test of residual error-disturbance uncertainty relations for mixed spin-1/2 states. Phys. Rev. Lett. 117, 140402 (2016)
Birkhoff and von Neumann [11] introduced quantum logic in response to logical problems with the Copenhagen interpretation. Takeuti [8] formed the quantum logic into an introduction of QST. Eventually, Ozawa took the lead to develop QST into a feasible interpretation of quantum mechanics (QM) that coheres with the classical law of noncontradiction, predicate logic, and experimental physics [9, 10, 12, 13]. For example, QST defines quantum states with certainty instead of classical uncertainty.
In short, QST begins with a prior probability distribution of observables for a particular quantum state. This distribution looks similar to a corresponding Copenhagen probability distribution of observables for the quantum state, but QST assigns predicate logic to each observable in the prior set for the quantum state. For example, the existence of each observable in a particular quantum state is true or false. This results in a set of existing observables that completely defines the quantum state despite classically non-commuting observables such as momentum and position. Additionally, QST can define entangled states because it is a state-dependent theory instead of a particle-dependent theory. Furthermore, QST preserves the Copenhagen probability distribution for wave function collapse that results in probabilistic causality during the transition from one quantum state to the next. For instance, wave function collapse is the only element of uncertainty in QST.
References
8. Takeuti, G.: Quantum set theory. In: Beltrametti, E.G., van Fraassen, B.C. (eds.) Current Issues in Quantum Logic, pp. 303–322. Plenum, New York (1981)
9. Ozawa, M. Quantum reality and measurement: A quantum logical approach. Found. Phys. 41, 592–607 (2011)
10. Ozawa, M. Quantum set theory extending the standard probabilistic interpretation of quantum theory. New Generat. Comput. 34, 125–152 (2016)
11. Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)
12. Sulyok, G., Sponar, S., Demirel, B., Buscemi, F., Hall, M.J.W., Ozawa, M., Hasegawa, Y.: Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements. Phys. Rev. Lett. 115, 030401 (2015)
13. Demirel, B., Sponar, S., Sulyok, G., Ozawa, M., Hasegawa, Y.: Experimental test of residual error-disturbance uncertainty relations for mixed spin-1/2 states. Phys. Rev. Lett. 117, 140402 (2016)
